Discrete OptimizationIterated tabu search for the circular open dimension problem
Highlights
► An effective iterated tabu search algorithm (ITS) is originally proposed for CODP. ► A robust tabu search procedure is implemented. ► A solution perturbation operator as well as an acceptance criterion is designed. ► ITS improves 76 best known results out of 146 instances within reasonable time.
Introduction
Cutting and packing (C&P) problems are widely encountered in practical applications, such as paper industry (Fraser and George, 1994), wireless communication (Adickes et al., 2002), marine transport (Birgin et al., 2005), aircraft designing (Liu and Li, 2010), and material cutting (He et al., 2012). They generally consist of cutting (or packing) a set of small items from (or into) a large object so as to minimize the wasted portion. As well-known NP-hard problems, C&P problems are extremely challenging to solve exactly and thus heuristics which attempt to obtain approximate solutions within reasonable time have been the most proposed approaches for tackling C&P problems.
As a representative variant of the C&P family, the circle packing problem (denoted by CPP) is concerned about how to pack a number of circles of known radii into a larger container without overlapping. The objective is to minimize the container size. According to the size of the circles to pack, CPP can be classified into two categories (Castillo et al., 2008): the arbitrary sized circle packing problem (denoted by ACP) and the uniform sized circle packing problem (denoted by UCP). It should be pointed out that ACP has to deal with both continuous and combinatorial features of the problem, while UCP mainly deals with continuous optimization problem. Due to the extremely large-scale combinatorial solution space of ACP, the approaches proposed for ACP are usually very different from those proposed for UCP.
This paper investigates a variant of ACP: the circular open dimension problem (denoted by CODP), which attempts to pack N(N = 1, 2, …) arbitrary sized circles into a strip of fixed width and unlimited length without overlapping. More precisely, CODP can be formulated as follows.
Given a strip of fixed width W and unlimited length L, as well as N arbitrary sized circles Ci of known radii ri(i = 1, 2, … , N). Take the origin of two-dimensional Cartesian coordinate system at the midpoint of the container, and denote the coordinates of the midpoint of Ci by (xi, yi). The objective of CODP is to obtain a solution (X, L), where X is a configuration denoted by (x1, y1, … , xi, yi, … , xN, yN), such that.
Minimize L, subject to:
The first two constraints state that each circle should not extend outside the container. The third constraint requires that any pair-wise circles cannot overlap with each other. (X, L) is termed a feasible solution if it meets all the constraints.
Furthermore, in order to measure the feasibility of a given solution (X, L), we define a penalty function based on the definition of overlaps as follows. For any solution (X, L), there may exist two kinds of overlaps: overlaps between two circles and overlaps between a circle and a border of the strip. Respectively, the overlapping depth between the ith circle Ci and the jth circle Cj is
And the overlapping depth between Ci and a vertical border of the strip is
Similarly, the overlapping depth between Ci and a horizontal border of the strip is
By adding all squares of overlapping depth together, we get a penalty function E(X, L) which measures the feasibility of a solution (X, L) as follows
According to this definition, it is not difficult to find that: (1) For any solution (X, L), E(X, L) ⩾ 0. (2) (X, L) is feasible if and only if E(X, L) = 0. Therefore, the objective of CODP is to minimize L while guaranteeing E(X, L) = 0.
In the present work, we try to solve CODP by a series of sub-problems, each with a fixed strip length. For each sub-problem, an iterated tabu search approach named ITS is proposed, which is composed of a tabu search procedure and a perturbation operator, associated with an acceptance criterion. As supplementary methods, some post-processing techniques are implemented to determine the strip length in monotonously decreasing mode. Computational results demonstrate that ITS is rather competitive with respect to the state-of-the-art approaches, in terms of both solution quality and computation time.
The rest of this paper is organized as follows: Section 2 briefly reviews the relevant literature. Section 3 presents the details of the proposed ITS algorithm. Computational experiments and analysis are presented in Sections 4 Computational results, 5 Conclusion concludes the paper.
Section snippets
Related literature
Various approaches have been developed for solving the circle packing problems (CPP). In this section, we briefly review the approaches proposed for solving CPP, especially for the circular open dimension problem (CODP) and its closely related variants.
As mentioned above, CPP can be classified into two categories: the arbitrary sized circle packing problem (ACP) and the uniform sized circle packing problem (UCP). Herein, we introduce several representative approaches proposed for ACP at first
Proposed approach
This paper studies CODP, which considers how to pack a number of arbitrary sized circles into a strip of fixed width and unlimited length without overlapping. In this paper, the original optimization problem is solved by a series of sub-problems, each considers about how to feasibly pack all the circles into a rectangular container of fixed width and fixed length. As supplementary techniques, we try to determine the strip length in a monotonously decreasing mode, just as detailed as follows.
- (1)
Computational results
In order to evaluate the performance of the proposed approach, we implement ITS (as detailed in Algorithm 1) in C++ language and run it on several computers, each with an Intel Xeon E5440 2.83 gigahertz processor and 2 gigabytes RAM. Note that when doing the computational experiments, each benchmark instance occupies only one processor.
Two sets of 146 representative instances, which have been widely used as benchmarks by previous researchers, are taken from the literature and tested.
Conclusion
Cutting and packing (C&P) problems are well known NP-hard problems and are widely encountered in practical applications. This paper investigates the circular open dimension problem (CODP), which is a representative variant of the C&P family. For this problem, an iterated tabu search algorithm named ITS is proposed, which is composed of a tabu search procedure (TS) and a solution perturbation operator associated with an acceptance criterion. As a representative perturbation-based approach, the
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 61173180, 61100144, 61100076 and 71201119). Sincere thanks to the reviewers for their professional comments and suggestions which helped us to improve the quality of this paper significantly. Bo Peng also provided a lot of help.
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